Quantum phyics project for a high schooler 
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Study Quantum Physics 

I am a high schooler who is interested in physics and mathematics, and I have a kind of 'high-school thesis' coming up in a year and a half or so. I want mine to be about Quantum Physics, and I have already prepared by self-studying Linear Algebra (and I'm planning on starting with Differential Equations too). I just have a couple of obstacles:


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*If I know Linear Algebra and (Ordinary & Partial) Differential Equations, what  parts of Quantum Physics will be in my reach?

*What other fields of mathematics would you recommend me studying to gain a grasp of rudimentary quantum physics and to make me able to make a  thesis on quantum physics?

*What would you guys consider interesting for a thesis on quantum physics? What are fascinating experiments, concepts, results, etc. of quantum physics that would be worth making a thesis about?
 A: Here is a nice thing you can do at a high school level--- show if it is possible to take a photograph in complete darkness.
Suppose you have a dark room, and you don't want to disturb it, you don't want to shine light on it. But you want to see what's in the room. You have access to a light source, a window into the room (or you are in the room), beam-splitters, and interferometers, but you want to ensure that at any time the probability of a photon getting detected inside the room is negligible. Can you take a picture?
Classically you can't do it, you either shine a photon into the room, or you don't. But, with a modification of the Elitzur-Vaidman bomb tester, you should be able to easily build a theoretical device which can scan a photograph of the room without ever allowing an appreciable probability of a photon ever being detected inside the room, outside your apparatus.
The mechanism has not been worked out in full, this is an old original idea of mine, but it is simple given the Elitzur Vaidman result. A sketch is as follows--- you do the Elitzur Vaidman thing, split a photon into N components with magnitude $\epsilon$, where $N\epsilon^2$ is small, and scatter all these little photon components off the room in separated bunches. You then have a rotating lens which collects the photon amplitude fraction that come back out the window, and interferes with the major component of the beam. You can detect interference when $N\epsilon$ is non-negligible, but this still makes it that a person inside the room will detect no photons. You can make the illumination going into the room arbitraily small, while allowing the interference effects to be detectable, and to reveal at which angles the photon returns to you, or if it is absorbed.
I do not know if this can be used to make a practical dark camera, but it is possible in principle, and it would make a nice high-school project which I think would be guaranteed a top prize, even if it is purely theoretical. It's another counterintuitive property of quantum mechanics, the counterfactual measurement.
A: I think most people would agree that Bell's theorem is one of the most fascinating and perplexing results in quantum theory. In short, Bell's theorem quantifies the non-separability of quantum mechanics in precise terms. Non-separability means that, contrary to intuition, it is sometimes not possible to talk about physical properties (such as momentum, position etc.) belonging to one particle alone. 
Even if two electrons are a million light-years apart, if they are in an entangled state (which will generally be true if they interacted in the past) then the outcomes of measurements on the two particles are correlated. Bell showed that these correlations are such that the measurement outcomes cannot be explained by properties belonging to one particle or the other, rather you can only speak about properties belonging to the system as a whole (in this case the system means the pair of electrons together). This strikes me (and a lot of other people too, notably Einstein) as pretty weird, since you wouldn't expect the momentum of electron A that is sitting here in your lab on Earth to depend on electron B, which could be a million light years away!
Alternatively, Bell's theorem shows that if you want to explain the outcomes of your measurements in terms of properties belonging just to one electron or the other, then the electrons must be able to communicate instantaneously over arbitrarily large distances, which does not sit well with our understanding of other laws of physics like relativity. The amazing thing is that Bell's theoretical result was proved correct by experiments by Alain Aspect in the 80's (if I remember right).
Hopefully that convinces you that Bell's theorem is interesting. It is a suitable topic for your project because the maths required is minimal: you already know more than enough to understand it (although perhaps learning a bit of elementary probability theory would help in reading Bell's original papers). That does not mean that the physics won't make your head hurt though :) There are also lots of other related things you could talk about: the experiments that confirmed Bell's theorem, the relationship of entanglement to quantum computing etc. 
You can find out more by reading Bell's book "Speakable and Unspeakable in Quantum Mechanics", which anyway should be required reading for any budding student of quantum mechanics.
